chapter 1 safa kasap university of saskatchewan canada · 2017-12-04 · e = pe= ×Š = Š 4.23 ev...

Solutions to Principles of Electronic Materials and Devices: 4th Edition (25 April 2017) Chapter 1

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Solutions Manual to Principles of Electronic Materials and Devices

Fourth Edition © 2018 McGraw-Hill

CHAPTER 1

Safa Kasap

University of Saskatchewan Canada

Check author's website for updates http://electronicmaterials.usask.ca

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Water molecules are polar. A water jet can be bent by bringing a charged comb near the jet. The polar

molecules are attracted towards higher fields at the comb's surface (Photo by SK)

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Fourth Edition (© 2018 McGraw-Hill)

Chapter 1

Answers to "Why?" in the text

Page 31:

Oxygen has an atomic mass of 16 whereas it is 14 for nitrogen. The O2 molecule is therefore heavier than

the N2 molecule. Thus, from )(3 212

21 kTm =v , the rms velocity of O2 molecules is smaller than that of N2

molecules.

Page 34, footnote 11

For small extensions, the difference between the engineering and instantaneous strains due to a temperature change are the same. Historically, mechanical and civil engineers measured extension by monitoring the change in length, ΔL; and the instantaneous length L was not measured. It is not trivial to measure both the instantaneous length and the extension simultaneously. However, since we know Lo and measure ΔL, the instantons length L = Lo + ΔL. Is the difference important? Consider a sample of length Lo that extends to a final length L due to a temperature change from To to T. Let ε = (L − Lo) / Lo = ΔL/Lo be the engineering strain.

The engineering definition of strain and hence the thermal expansion coefficient is

Engineering strain = TL

L

o

λδδ =

so that thermal expansion from To to T gives,

=T

T

L

L o oo

dTL

dL λ ∴ )( oo

TTL

L −=Δ λ ∴ )( oTT −= λε (1)

where ε = ΔL/Lo is the engineering strain as defined above

Physics definition of strain and hence the thermal expansion coefficient is

Instantaneous train = TL

L λδδ =

so that thermal expansion from To to T gives,

=T

T

L

L oo

dTL

dL λ ∴ )(ln oo

TTL

L −=

λ ∴ ( ) )(1ln oTT −=+ λε (2)

We can expand the ln(1 + ε) term for small ε, so that Equation (2) essentially becomes Equation (1)

1.1 Virial theorem The Li atom has a nucleus with a +3e positive charge, which is surrounded by a full 1s shell with two electrons, and a single valence electron in the outer 2s subshell. The atomic radius of the Li atom is about 0.17 nm. Using the Virial theorem, and assuming that the valence electron sees the






nuclear +3e shielded by the two 1s electrons, that is, a net charge of +e, estimate the ionization energy of Li (the energy required to free the 2s electron). Compare this value with the experimental value of 5.39 eV. Suppose that the actual nuclear charge seen by the valence electron is not +e but a little higher, say +1.25e, due to the imperfect shielding provided by the closed 1s shell. What would be the new ionization energy? What is your conclusion?

Solution

First we consider the case when the outermost valence electron can see a net charge of +e. From Coulomb’s law we have the potential energy

0000 rπε

ee

rπε

QQPE

4

))((

421 −+==

m)10)(0.17Fm108.85(4

C)10(1.69112

219

−−−

−

×××−=

π = −1.354 × 10−18 J or −8.46 eV

Virial theorem relates the overall energy, the average kinetic energy KE , and average potential energy PE through the relations

KEPEE += and PEKE2

1−=

Thus using Virial theorem, the total energy is

eV46.85.02

1 −×== PEE = − 4.23 eV

The ionization energy is therefore 4.23 eV.

Consider now the second case where the electron sees +1.25e due to imperfect shielding. Again the Coulombic PE between +e and +1.25e will be

0000

21

r4π

ee

r4π

QQPE

εε))(1.25( −+==

m)10)(0.17Fm10(854

C)10(1.61.259112

219

−−−

−

×××⋅−=

π= −1.692 × 10−18 J or −10.58 eV

The total energy is,

eV29.52

1 −== PEE

The ionization energy, considering imperfect shielding, is 5.29 eV. This value is in closer agreement with the experimental value. Hence the second assumption seems to be more realistic.

1.2 Virial theorem and the He atom In Example 1.1 we calculated the radius of the H-atom using the Virial theorem. First consider the He+ atom, which as shown in Figure 1.75a, has one electron in the K-sell orbiting the nucleus. Take the PE and the KE as zero when the electrons and the nucleus are infinitely separated. The nucleus has a charge of +2e and there is one electron orbiting the nucleus at a radius r2. Using the Virial theorem show that the energy of the He+ ion is






2

2

4

2)2/1()He(

r

eE

oπε−=+ Energy of He+ ion [1.48]

Now consider the He-atom shown in Figure 1.75b. There are two electrons. Each electron interacts with the nucleus (at a distance r1) and the other electron (at a distance 2r1). Using the Virial theorem show that the energy of the He atom is

−=

1

2

8

7)2/1()He(

r

eE

oπε Energy of He atom [1.49]

The first ionization energy EI1 is defined as the energy required to remove one electron from the He atom. The second ionization energy EI2 is the energy required to remove the second (last) electron from He+. Both are shown in Figure 1.75 These have been measured and given as EI1 = 2372 kJ mole−1 and EI2 = 5250 kJ mol−1. Find the radii r1 and r2 for He and He+. Note that the first ionization energy provides sufficient energy to take He to He+, that is, He → He+ + e− absorbs 2372 kJ mol−1. How does your r1 value compare with the often quoted He radius of 31 pm?

Figure 1.75: (a) A classical view of a He+ ion. There is one electron in the K-shell orbiting the nucleus that has a charge +2e. (b) The He atom. There are two electrons in the K-shell. Due to their mutual repulsion, they orbit to

void each other. Solution

Virial theorem relates the overall energy, the average kinetic energy KE , and average potential energy PE through the relations

EKEPEEPEKEKEPEE2

1;

2

1;

2

1; −==−=+= (1)

Now, consider the PE of the electron in Figure 1.75a. The electron interacts with +2e of positive charge, so that

2

2

2 4

2

4

)2)((

r

e

r

eePE

oo πεπε−=−=

which means that the total energy (average) is

2

2

2

2

44

2)2/1(

2

1)He(

r

e

r

ePEE

oo πεπε−=−==+ (2)

which is the desired result.






Now consider Figure 1.75b. Assume that, at all times, the electrons avoid each other by staying in opposite parts of the orbit they share. They are "diagonally" opposite to each other. The PE of this system of 2 electrons one nucleus with +2e is PE = PE of electron 1 (left) interacting with the nucleus (+2e), at a distance r1 + PE of electron 2 (right) interacting with the nucleus (+2e), at a distance r1 + PE of electron 1 (left) interacting with electron 2 (right) separated by 2r1

∴ )2(4

))((

4

)2)((

4

)2)((

111 r

ee

r

ee

r

eePE

ooo πεπεπε−−+−+−=

∴ 2

2

8

7

r

ePE

oπε−=

From the Virial theorem in Equation (1)

−=

1

2

8

7)2/1()He(

r

eE

oπε (3)

We are given, EI1 = Energy required to remove one electron from the He atom = 2372 kJ mole−1 = 25.58 eV EI2 = Energy required to remove the second (last) electron from He+ = 5250 kJ mol−1 = 54.41 eV The eV values were obtained by using

=

AeNEE

1)J/mole()eV(

We can now calculate the radii as follows. Starting with Equation 2 for the ionization of He+,

2112

219

2

2

219

2 )mF10854.8(4

)C10602.1(

4)He()J/eV10602.1)(eV41.54(

rr

eEEE

oII −−

−+−

××====×=

ππε

from which,

r2 = 2.65×10−11 or 26.5 pm The calculation of r1 involves realizing that Equation (3) is the energy of the whole He atom, with 2 electrons. If we remove 1 electron we are left with He+ whose energy is Equation (2). Thus the

)He()He(1+−= EEEI

∴ 21

219

8

7)2/1()He()He(J/eV)10602.1(eV)58.24( I

o

Er

eEE −

=−=× +−

πε

∴ 1

112

219

1

219

)mF10854.8(4

)C10602.1(

8

7)2/1()J/eV10602.1(eV)58.24eV41.54(

rr

e

o−−

−−

××=

=×+

ππε

from

r1 = 3.19×10−11 or 31.9 pm very close to the quoted value of 31 pm in various handbooks or internet period tables ____________________________________________________________________________________

1.3 Atomic mass and molar fractions






a. Consider a multicomponent alloy containing N elements. If w1, w2, ..., wN are the weight fractions of components 1,2,..., N in the alloy and M1, M2, ..., MN, are the respective atomic masses of the elements, show that the atomic fraction of the i-th component is given by

N

N

iii

M

w

M

w

M

wMw

n+++

=...

/

2

2

1

1

Weight to atomic percentage

b. Suppose that a substance (compound or an alloy) is composed of N elements, A, B, C,... and that we know their atomic (or molar) fractions nA, nB nC, .... Show that the weight fractions wA, wB, wC,....are given by

...+++

=CCBBAA

AAA MnMnMn

Mnw

...+++=

CCBBAA

BBB MnMnMn

Mnw Atomic to weight percentage

c. Consider the semiconducting II-VI compound cadmium selenide, CdSe. Given the atomic masses of Cd and Se, find the weight fractions of Cd and Se in the compound and grams of Cd and Se needed to make 100 grams of CdSe.

d. A Se-Te-P glass alloy has the composition 77 wt.% Se, 20 wt.% Te and 3 wt.% P. Given their atomic masses, what are the atomic fractions of these constituents?

Solution

a. Suppose that n1, n2, n3,…, ni,…, nN are the atomic fractions of the elements in the alloy,

n1 + n2 + n3 +… + nN = 1

Suppose that we have 1 mole of the alloy. Then it has ni moles of an atom with atomic mass Mi (atomic fractions also represent molar fractions in the alloy). Suppose that we have 1 gram of the alloy. Since wi is the weight fraction of the i-th atom, wi is also the mass of i-th element in grams in the alloy. The number of moles in the alloy is then wi/Mi. Thus,

Number of moles of element i = wi/Mi

Number of moles in the whole alloy = w1/M1 + w2/M2 +…+ wi/Mi +…+wN/MN

Molar fraction or the atomic fraction of the i-th elements is therefore,

alloyinmolesofnumbersTotal

elementofmolesofNumebr ini =

∴

N

N

iii

M

w

M

w

M

wMw

n+++

=...

/

2

2

1

1

b. Suppose that we have the atomic fraction ni of an element with atomic mass Mi. The mass of the element in the alloy will be the product of the atomic mass with the atomic fraction, i.e. niMi. Mass of the alloy is therefore

nAMA + nBMB + … + nNMN = Malloy By definition, the weight fraction is, wi = mass of the element i/Mass of alloy. Therefore,

...+++

=CCBBAA

AAA MnMnMn

Mnw






...+++

=CCBBAA

BBB MnMnMn

Mnw

c. The atomic mass of Cd and Se are 112.41 g mol−1 and 78.96 g mol−1. Since one atom of each element is in the compound CdSe, the atomic fraction, nCd and nSe are 0.5. The weight fraction of Cd in CdSe is therefore

11

1

SeSeCdCd

CdCdCd molg96.785.0molg41.1125.0

molg41.1125.0−−

−

×+××=

+=

MnMn

Mnw = 0.587 or 58.7%

Similarly weight fraction of Se is

11

1

SeSeCdCd

SeSeSe molg96.785.0molg41.1125.0

molg96.785.0−−

−

×+××=

+=

MnMn

Mnw = 0.4126 or 41.3%

Consider 100 g of CdSe. Then the mass of Cd we need is Mass of Cd = wCdMcompound = 0.587 × 100 g = 58.7 g (Cd) and Mass of Se = wSeMcompound = 0.413 × 100 g = 41.3 g (Se) d. The atomic fractions of the constituents can be calculated using the relations proved above. The atomic masses of the components are MSe = 78.6 g mol−1, MTe = 127.6 g mol−1, and MP = 30.974 g mol−1. Applying the weight to atomic fraction conversion equation derived in part (a) we find,

111

1

P

P

Te

Te

Se

Se

SeSeSe

molg974.30

03.0

molg6.127

2.0

molg6.78

77.0molg6.78

77.0

/

−−−

−

++=

++=

M

w

M

w

M

wMw

n

∴ nSe = 0.794 or 79.4%

111

1

P

P

Te

Te

Se

Se

TeTeTe

molg974.30

03.0

molg6.127

2.0

molg6.78

77.0molg6.127

2.0

/

−−−

−

++=

++=

M

w

M

w

M

wMw

n

∴ nTe = 0.127 or 12.7 %

111

1

P

P

Te

Te

Se

Se

PPP

molg974.30

03.0

molg6.127

2.0

molg6.78

77.0molg974.30

03.0

/

−−−

−

++=

++=

M

w

M

w

M

wMw

n

∴ nP = 0.0785 or 7.9%

1.4 Mean atomic separation, surface concentration and density There are many instances where we only wish to use reasonable estimates for the mean separation between the host atoms in a crystal and the






mean separation between impurities in the crystal. These can be related in a simple way to the atomic concentration of the host atoms and atomic concentration of the impurity atoms respectively. The final result does not depend on the sample geometry or volume. Sometimes we need to know the number of atoms per unit area ns on the surface of a solid given the number of atoms per unit volume in the bulk, nb. Consider a crystal of the material of interest which is a cube of side L as shown in Figure 1.76. To each atom, we can attribute a portion of the whole volume, which is a cube of side a. Thus, each atom is considered to occupy a volume of a3. Suppose that there are N atoms in the volume L3. Thus, L3 = Na3.

a. If nb is the bulk concentration of atoms, show that the mean separation a between the atoms is given by 3/1 bna = .

b. Show that the surface concentration ns of atoms is given by 3/2bs nn = .

c. Show that the density of the solid is given by Ab NMn /at=ρ where Mat is the atomic mass.

Calculate the atomic concentration in Si from its density (2.33 g cm−3)

d. A silicon crystal has been doped with phosphorus. The P concentration in the crystal is 1016 cm−3. P atoms substitute for Si atoms and are randomly distributed in the crystal. What is the mean separation between the P atoms?

Figure 1.76 Consider a crystal that has volume L3. This volume is proportioned to each atom, which is a cube of

side a3. Solution

a. Consider a crystal of the material which is a cube of volume L3, so that each side has a length L as shown in Figure 1.76. To each atom, we can attribute a portion of the whole volume. For simplicity, we take the volume proportioned to an atom to be a3, that is, each atom is considered to occupy a volume of a3.

The actual or true volume of the atom does not matter. All we need to know is how much volume an atom has around it given all the atoms are identical and that adding all the atomic volumes must give the whole volume of the crystal.

Suppose that there are N atoms in this crystal. Then nb = N/L3 is the atomic concentration in the crystal, the number of atoms per unit volume, the so-called bulk concentration. Since N atoms make up the crystal, we have

Na 3 = Crystal volume = L3






The separationbetween any two atoms is a, as shown in Figure 1.76. Thus,

3/1

3/13/13 11separationMean

bb nnN

La =

=

== (1)

Equation 1 can be derived even more simply because an atom has a volume of a3. In this volume are is only 1 atom. So nba3 must be 1, which leads to Equation 1.

The above idea can be extended to finding the separation between impurity atoms in a crystal. Suppose that we wish to determine the separation d between impurities, then we can again follow a similar procedure. In this case, the atoms in Figure 1.76 are impurities which are separated by d. We again assign a portion of the whole volume (for simplicity, a cubic volume) to each impurity. Each impurity atom therefore has a volume d3 (because the separation between impurities is d). Following the above arguments we would find,

3/1

1impuritiesbetweenseparationMean

INd == (2)

b. We wish to find the number of atoms per unit area ns on the surface of a solid given the atomic concentration nb in the bulk. Consider Figure 1.76. Each atom as an area a2, so that within this surface area there is 1 atom. Thus

Surface area of 1 atom × Surface concentration of atoms = 1

or a2ns = 1

∴ 3/22

1bs n

an == (3)

Equation (3) is of course based on the simple arrangement of atoms as shown in Figure 1.75. In reality, the surface concentration of atoms depends on the crystal plane at the surface. Equation (3), however, is a reasonable estimate for the order of magnitude for ns given nb.

c. We can determine the density ρ from the atomic concentration nb and vice versa. The volume in Figure 1.75 has L3nb atoms. Thus, the density is

A

bAb

N

Mn

L

NM

nLat

3

at3 )(

Volume

Mass ===ρ (4)

For Si, the atomic mas Mat = 28.09 g mol−1, so that with ρ given as 2.33 g cm−3,

)molg09.28(

)mol10022.6)(cmg33.2(1

1233

at−

−− ×==M

Nn A

b

ρ= 5.00×1022 cm−3.

d. The P concentration in the crystal is 1016 cm−3. P atoms substitute for Si atoms and are randomly distributed in the crystal. We can use Equation (2)

3/136163/1 )m10101(

11impuritiesbetweenseparationMean −××

===IN

d

= 4.64×10−8 m = 46.4 nm






1.5 The covalent bond Consider the H2 molecule in a simple way as two touching H atoms as depicted in Figure 1.77. Does this arrangement have a lower energy than two separated H atoms? Suppose that electrons totally correlate their motions so that they move to avoid each other as in the snapshot in Figure 1.77. The radius ro of the hydrogen atom is 0.0529 nm. The electrostatic potential energy PE of two charges Q1 and Q2 separated by a distance r is given by Q1Q2/(4πεor). Using the Virial Theorem as in Example 1.1, consider the following:

a. Calculate the total electrostatic potential energy (PE) of all the charges when they are arranged as shown in Figure 1.77. In evaluating the PE of the whole collection of charges you must consider all pairs of charges and, at the same time, avoid double counting of interactions between the same pair of charges. The total PE is the sum of the following: electron 1 interacting with the proton at a distance ro on the left, proton at ro on the right, and electron 2 at a distance 2ro + electron 2 interacting with a proton at ro and another proton at 3ro + two protons, separated by 2ro, interacting with each other. Is this configuration energetically favorable?

b. Given that in the isolated H-atom the PE is 2 ×(−13.6 eV), calculate the change in PE in going from two isolated H-atoms to the H2 molecule. Using the Virial theorem, find the change in the total energy and hence the covalent bond energy. How does this compare with the experimental value of 4.51 eV?

Figure 1.77 A simplified view of the covalent bond in H2. A snapshot at one instant.

Solution

a. Consider the PE of the whole arrangement of charges shown in the figure. In evaluating the PE of all the charges, we must avoid double counting of interactions between the same pair of charges. The total PE is the sum of the following:

Electron 1 interacting with the proton at a distance ro on the left, with the proton at ro on the right and with electron 2 at a distance 2ro

+ Electron 2 on the far left interacting with a proton at ro and another proton at 3ro

+ Two protons, separated by 2ro, interacting with each other






PE = −e2

4πεoro

−e2

4πεoro

+e2

4πεo (2ro )

− e2

4πεoro

− e2

4πεo 3ro

+ e2

4πεo 2ro

Substituting and calculating, we find PE = 1.0176 × 10−17 J or -63.52 eV

The negative PE for this particular arrangement indicates that this arrangement of charges is indeed energetically favorable compared with all the charges infinitely separated (PE is then zero).

b. The potential energy of an isolated H-atom is -2× 13.6 eV or -27.2 eV. The difference between the PE of the H2 molecule and two isolated H-atoms is,

ΔPE = −(63.52) eV − 2(-27.2) eV = −9.12eV

We can write the last expression above as the change in the total energy.

eV56.4)eV12.9(2

1

2

1 −=−=Δ=Δ PEE

This change in the total energy is negative. The H2 molecule has lower energy than two H-atoms by 4.56 eV which is the bonding energy. This is very close to the experimental value of 4.51 eV. (Note: We used a ro value from quantum mechanics - so the calculation was not totally classical!).

1.6 Ionic bonding and CsCl The potential energy E per Cs+-Cl− pair within the CsCl crystal depends on the interionic separation r in the same fashion as in the NaCl crystal,

mo r

B

r

MerE +−=

πε4)(

2

Energy per ion pair in ionic crystals [1.48]

where for CsCl, M = 1.763, B = 1.192×10−104 J m9 or 7.442×10−5 eV (nm) 9 and m = 9. Find the equilibrium separation (ro) of the ions in the crystal and the ionic bonding energy, that is, the ionic cohesive energy; and compare the latter value to the experimental value of 657 kJ mol−1. Given the ionization energy of Cs is 3.89 eV and the electron affinity of Cl (energy released when an electron is added) is 3.61 eV, calculate the atomic cohesive energy of the CsCl crystal as joules per mole. Solution

Bonding will occur when potential energy E(r) is minimum at r = r0 corresponding to the equilibrium separation between Cs+ and Cl− ions. Thus, differentiating E(r) and setting it equal to zero at r = ro we have

04

)( 2

=

+−=

==oo rr

morr r

B

r

Me

dr

d

dr

rdE

πε

∴ 04 12

2

=

−

=+

orr

mo r

Bm

r

Me

πε






∴ 04 12

2

=− +mooo r

Bm

r

Me

πε

∴ 1

1

2

4 −

=

mo

o Me

mBr

πε (1)

Thus substituting the appropriate values we have

( )8

1

219

9104112

C106.1763.1

)mJ10192.1(9)Fm108542.8(4

××××××=

−

−−−πor

∴ ro = 3.57 × 10−10 m or 0.357 nm. The minimum energy is the energy at r = ro, that is

m

ooo r

B

r

MeE +−=

πε4

2

min

which in terms of eV is

9

9

min )nm(

)nmeV(

4)eV(

ooo r

B

r

eME +−=

πε (2)

9

94

10112

19

)nm357.0(

nmeV10442.7

)m1057.3)(Fm108542.8(4

)763.1)(C106.1( −

−−−

− ×+××

×−=π

= − 6.32 eV per ion pair, or 3.16 eV per ion.

The amount of energy required to break up a CsCl crystal into Cs+ and Cl− ions = 6.32 eV per pair of ions. The corresponding ionic cohesive energy is

Ecohesive = (6.32 eV)(1.602 × 10−19 J eV−1)(6.022× 1023 mol−1)

= 610 kJ mol─1 of Cs+Cl− ion pairs or 610 kJ mol─1 of Cs+ ions and Cl− ions. (Not far out from the experimental value given the large numbers and the high index, m = 9, involved in the calculations.) The amount of energy required to remove an electron from Cl− ion = 3.61 eV. The amount of energy released when an electron is put into the Cs+ ion = 3.89 eV. Bond Energy per pair of Cs-Cl atoms = 6.32 eV + 3.61 eV – 3.89 eV = 6.04 eV Atomic cohesive energy in kJ/mol is,

Ecohesive = (6.04 eV)(1.6 × 10−19 J eV−1)(6.022× 1023 mol−1) = 582 kJ mol─1 of Cs or Cl atom (i.e. per mole of Cs-Cl atom pairs) = 291 kJ mol─1 of atoms ------------------------ Notes: 1. Various books and articles report different values for B and m for CsCl, which obviously affect the calculated energy; ro is less affected because it requires the (m−1)th root of mB. Richard Christman






(Introduction to Solid State Physics, John Wiley and Sons, 1988) in Table 5-1 gives, m = 10.65 and B = 3.44 × 10120, quite different than values here, which are closer to values in Alan Walton, Three Phases of Matter (2nd Edition), Clarendon (Oxford University) Press, 1983 (pp. 258-259). 2. The experimental value of 657 kJ mol−1 (6.81 eV per ion pair) for the ionic cohesive energy (the ionic lattice energy) is from T. Moeller et al, Chemistry with Inorganic Qualitative Analysis, Second Edition, Academic Press, 1984) p. 413, Table 13.5. 3. Some authors use the term molecular cohesive energy to indicate that the crystal is taken apart to molecular units e.g. Cs+Cl−, which would correspond to the ionic cohesive energy here. Further, most chemists use "energy per mole" to imply energy per mole of chemical units, and hence the atomic cohesive energy per mole would usually refer to energy be per Cs and Cl atom pairs. Some authors refer to the atomic cohesive energy per mole as cohesive energy per mole of atoms, independent of chemical formula. 4. Equation (1) is

12

4−

= m

oo

rm

MeB

πε

which can be substituted into Equation (2) or [1.48] to eliminate B, that is, find the minimum energy of a pair of Cs+-Cl− ions, i.e.

122

min 4

1

4−

+−= m

oo

mo

rm

Me

rr

MeE

πεπε

∴

−−=

mr

MeE

oo

11

4

2

min πε

which is called the Born-Landé equation.

1.7 Ionic bonding and LiCl Equation 1.48 can be used to represent the PE of the ion pair inside the LiC crystal. LiCl has the NaCl structure with M = 1.748, m = 7.30, B = 2.34×10−89 J m7.30. Further, the ionization energy of Li (Li → Li+ + e−) is 520.2 kJ mol−1. The electron affinity of Cl (energy associated with Cl + e− → Cl−) is 348.7 kJ mol−1 (a) Calculate the equilibrium separation of ions the LiCl crystal. (b) Calculate the bonding energy per ion pair in the LiCl crystal. (c) Calculate the atomic cohesive energy of the LiCl crystal. (c) Calculate the density of LiCl.

Solution

Figure 1Q07-1 shows the crystal structure of NaCl. LiCl has the same crystal structure.

(a) The minimum in energy is at

04

)( 2

=

+−=

==oo rr

morr r

B

r

Me

dr

d

dr

rdE

πε

∴ 04 12

2

=

−

=+

orr

mo r

Bm

r

Me

πε






∴ 04 12

2

=− +mooo r

Bm

r

Me

πε

∴ 1

1

2

4 −

=

mo

o Me

mBr

πε

Thus, substituting the appropriate values we have

8

1

219

30.789112

C)10602.1(748.1

)mJ1034.2(30.7)Fm108542.8(4

××

××××= −

−−−πor

∴ ro = 2.612× 10−10 m or 0.261 nm. (b) The minimum energy is the energy at r = ro, that is

m

ooo r

B

r

MeE +−=

πε4

2

min

8.710

8.789

10112

219

)m1061.2(

mJ1034.2

)m1061.2)(Fm108542.8(4

)748.1()C10602.1(−

−

−−−

−

××+

×××−=

π

= −1.33× 10−18 J per of Li+-Cl− ions

= −8.32 eV per ion pair, or −4.18 eV per ion.

The bond energy is therefore

Ebond = −Emin = 8.317 eV per ion pair, or 4.18 eV per ion.

(c) In (b) we calculated the amount of energy required to break up Li+-Cl− pair into Li+ and Cl− ions = 8.317 eV per pair of ions. The corresponding ionic cohesive energy is

Eionic-cohesive = (8.317 eV)(1.602 × 10−19 J eV−1)(6.022× 10−23 mol−1)

= 802 kJ mol─1 of Li+Cl− ion pairs or 802 kJ mol─1 of Li+ ions and Cl− ions. Consider the electron affinity of Cl and the ionization energy of Li, then by definition, we have

Energy required to remove an electron from Cl− ion = 348.7 kJ mol−1 = 3.614 eV per ion.

Energy released when an electron is put into the Li+ ion = 520.2 kJ mol−1 = 5.392 eV per ion. Bond Energy per pair of Cs-Cl atoms = 8.317 eV + 3.614 eV – 5.392 eV = 6.539 eV or 6.54 eV Atomic cohesive energy in kJ/mol is,

Eatomic-cohesive = (6.539 eV)(1.602 × 10−19 J eV−1)(6.022×1023 mol−1) = 631 kJ mol─1 of Cs or Cl atom (i.e. per mole of Cs-Cl atom pairs) = 315 kJ mol─1 of atoms (d) We know the interionic separation, which is ro. The lattice parameter a is

a = 2ro = 2(2.61×10−10 m) = 5.22×10−10 m






We can find the atomic masses of Li and Cl from Appendix C or from any periodic table. The density is given as

( )

3ClLi 44

unitcellofVolume

type thisofmassAtomiccellunitingiven type ofatomsofNumber

a

MM +=

×= ρ

∴ 310

123

13

123

13

)10224.5(

)mol10022.6()molkg1045.35(

4)mol10022.6()molkg1094.6(

4

−

−

−−

−

−−

××

×+×

×

=ρ = 1,980 kg m−3 or 1.98 g cm−3.

Figure 1Q07-1 Left: A schematic illustration of a cross section from solid NaCl. Solid NaCl is made of Cl− and Na+ ions arranged alternatingly, so the oppositely charged ions are closest to each other and attract each other.

There are also repulsive forces between the like-ions. In equilibrium, the net force acting on any ion is zero. The interionic separation ro and the lattice parameter a are shown; clearly a = 2ro. (b) Solid NaCl and the definition of

the unit cell with a lattice parameter a. There are 4 Na+ and 4 Cl− ions in the unit cell. Clearly, a = 2ro. ------------------------ Comments: The values for m, and B are from Richard Christman, Fundamental of Solid State Physics, Wiley and Sons (New York), 1987, Ch 5, Table 5-1, p130. Cl electron affinity from https://en.wikipedia.org/wiki/Electron_affinity_(data_page) and Li ionization energy from https://en.wikipedia.org/wiki/Molar_ionization_energies_of_the_elements (22 October 2016)

We can compare the results from this calculation with some published experimental and calculated values as in table 1Q07-1. The calculated values above are not too far out from experimental values. Usually m and B are determined by using experimental values for the elastic coefficients as will be apparent in Question 1.9 where elastic coefficients are related to m and B)

Lattice energy of an ionic crystal is the energy of formation of the crystal from its ions.

Ionic cohesive

kJ mol─1

Atomic cohesive per Li-Cl pair

kJ mol─1

Atomic cohesive energy per atom

kJ mol─1

ro

(nm)

a

(nm)

ρ

(g cm−3)

1Q07 (This question) 802 631 315 0.261 0.522 1.98






Wikipedia 0.514 2.068

Kittel, 7Ed, p73, Table 7 832.6 0.257

Moeller 834 0.257

Ashcroft and Mermin, Table 20.5, p406

831.1

C. Kittel, Silid State Physics, 7th Edition, John Wiley and Sons, New York, 1996; p73, Table 7. T. Moeller et al, Chemistry with Inorganic Qualitative Analysis, Second Edition, Academic Press, 1984) p. 413, Table 13.5. N.W. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College, 1976; Table 20.5 (p406) data, originally from M.P. Tossi, Solid State Physics, Vol. 16 (Edited by F. Seitz and D. Turnbull), Academic Press, New York, 1964, p54

1.8 Madelung constant If we were to examine the NaCl crystal in three dimensions, we would find that each Na+ ion has

6 Cl− ions as nearest neighbors at a distance r

12 Na+ ions as second nearest neighbors at a distance 2r

8 Cl− ions as third nearest neighbors at a distance 3r

and so on. Show that the electrostatic potential energy of the Na+ atom can be written as

r

Me

r

erE

oo πεπε 4...

3

8

2

126

4)(

22

−=

−+−−= Madelung constant M for NaCl

where M, called Madelung constant, is given by the summation in the square brackets for this particular ionic crystal structure (NaCl). Calculate M for the first three terms and compare it with M = 1.7476, its value had we included the higher terms. What is your conclusion? Solution

From Coulomb’s law of electrostatic attraction we know that the PE between two charges Q1 and Q2 separated by a distance r is given by

r

QQPE

oπε421=

First we consider the interaction between Na+ ion and 6Cl− ions at distance r. Applying Coulomb’s law we have

r

e

r

ee

r

QQPE

ooo πεπεπε 4

6

4

))(6(

4

221

1

−=+−==

Similarly, we now consider 12 Na+ ions as second nearest neighbors at a distance 2r

24

12

24

))(12(

4

221

2r

e

r

ee

r

QQPE

ooo πεπεπε=++==

and 8 Cl─ ions as third nearest neighbors at a distance 3r






34

8

34

))(8(

4

221

3r

e

r

ee

r

QQPE

ooo πεπεπε−=+−==

and similarly we can consider the next nearest set of neighbors and so on. Therefore, the overall PE of the Na+ ion is

...34

8

24

12

4

6)(

222

+−+−=r

e

r

e

r

erE

ooo πεπεπε

or r

Me

r

erE

oo πεπε 4...

3

8

2

126

4)(

22 −=

−+−−=

where clearly ⋅⋅⋅++−=3

8

2

126M

Considering just the first three terms we have M = 2.133. This is considerably different from the value M = 1.7464, the value obtained when higher order terms are considered. This implies that the next nearest neighbors have substantial effect on the potential energy. Note: See Appendix A for the explanation on how the PE of ions in the crystal is usually written. *1.9 Bonding and bulk modulus In general, the potential energy E per atom, or per ion pair, in a crystal as a function of interatomic (interionic) separation r can be written as the sum of an attractive PE and a repulsive PE,

E(r) = −A

r n +B

r m General PE curve for bonding [1.49]

where A and n are constants characterizing the attractive PE and B and m are constants characterizing the repulsive PE. This energy is minimum when the crystal is in equilibrium. The magnitude of the minimum energy and its location ro define the bonding energy and the equilibrium interatomic (or interionic) separation respectively.

When a pressure P is applied to a solid, its original volume Vo shrinks to V by an amount ΔV = V − V0. The bulk modulus K relates the volume strain ΔV/V to the applied pressure P by

P = −K(ΔV/Vo) Bulk modulus definition [1.50]

The bulk modulus K is related to the energy curve. In its simplest form (assuming a simple cubic unit cell) K can be estimated from Equation 1.50 by

orro dr

Ed

crK

=

=

2

2

9

1 Bulk modulus [1.51]

where c is a numerical factor, of the order of unity, given by b/p where p is the number of atoms or ion pairs in the unit cell and b is a numerical factor that relates the cubic unit cell lattice parameter ao to the equilibrium interatomic (interionic) separation ro by b = ao

3 / ro3

a. Show that the bond energy and equilibrium separation are given by

−=

m

n

r

AE

no

1bond and nm

o nA

mBr

−

=

1






b. Show that the bulk modulus is given by

39

)(+

−=n

ocr

nmAnK or 3bond 9 ocr

mnEK =

c. For a NaCl type crystal, Na+ and Cl− ions touch along cube edge so that ro = (ao/2). Thus, a3 = 23ro3

and b = 23 = 8. There are 4 ion pairs in the unit cell, p = 4. Thus, c = b/p = 8/4 = 2. Using the values from Example 1.3, calculate the bulk modulus of NaCl.

Solution

a. Interatomic separation r = ro is the distance at minimum E(r), Therefore we differentiate E(r) and set it equal to zero. i.e.

0)( =

+−=

== oo rrmn

rr r

B

r

A

dr

d

dr

rdE

∴ 011

=

−

=++

orrmn r

mB

r

An

∴ 011 =− ++ mo

no r

mB

r

An

∴ nA

mB

r

rn

o

mo =+

+

1

1

or nA

mBr nm

o =−

∴ nm

o nA

mBr

−

=

1

(1)

The potential energy is minimum at r = ro and is related with bonding energy E(ro) = −Ebond. From the equation for ro we have

An

Bmr nm

o =−

and isolate for B,

m

AnrB

nmo

−

= (2)

Substitute for B in the energy relation

+−=+−=

−

m

Anr

rr

A

r

B

r

ArE

nmo

mo

no

mo

no

o

1)(

no

no

n

no

mnm

no mr

An

r

A

m

Anr

r

A

m

Anr

r

A +−=+−=+−=−−−

00

∴

−−−=−=

m

n

r

ArEE

no

o 1)(bond (3)






∴

−=

m

n

r

AE

no

1bond (4)

b. Show that the bulk modulus is given by

K =An

9cron +3 m − n( ) or K =

mnEbond

9cro3

From the definition of Bulk modulus mentioned in the problem statement above

0

2

2

09

1

rrdr

Ed

crK

=

=

First, we find 2

2

dr

Ed, i.e.

+−=

mn r

B

r

A

dr

d

dr

rdE )(

11 ++ −=mn r

mB

r

An

∴ 0

112

2 )(

rrmn

rrr

mB

r

An

dr

d

dr

rdE

o =++

=

−=

2222

)1()1()1()1(

0

++=

++

+++−=

+++−= m

on

orrmn r

Bmm

r

Ann

r

Bmm

r

Ann

Again, substituting the value of B in the above relation, i.e. m

AnrB

nmo

−

= we have

nmmo

no

nm

mo

norr r

mAn

r

Ann

m

Anr

r

mm

r

Ann

dr

rdE

o

+−++

−

++=

+++−=+++−=

22

0222

2 )1()1()1()1()(

∴ ( )11)1()1()(

2222

2

++−−=+++−=

+++

=

mnr

An

r

mAn

r

Ann

dr

rdEn

on

on

orr o

∴ ( )nmr

An

dr

rdEn

orr o

−=

+

=22

2 )(

Now substitute for the second derivative in the equation for the Bulk modulus

−=

= +

=22

2 )(

9

1

9

1

0

noorro r

nmAn

crdr

Ed

crK

or 39

)(+

−=n

ocr

nmAnK (5)

From the relationship for the bonding energy,

mr

nmA

m

n

r

AE

no

no

)(1bond

−−=

−−=






Now, consider the expression for K and rearrange it as

39

)(

on

o cr

mn

mr

nmAK

−=

i.e. 3bond 9 ocr

mnEK =

c. From Example 1.3, the bonding energy for NaCl is M = 1.748, n = 1, m = 8, ro = 0.281 × 10−9 m, c = 2. Therefore,

)mF1085.8(4

)748.1()C106.1(

4 112

2192

−−

−

××==

ππεo

MeA = 4.022 × 10−28.

Substitute the above value for A in the expression for K in Equation (5),

319

28

3 )10281.0)(2(9

)18)(1)(10022.4(

9

)(+−

−

+ ×⋅−×=−=

nocr

nmAnK = 25.1 × 109 Pa or 25.1 GPa

------------------------ Note: Experimental value is roughly 2.4×1010 Pa or 24 GPa. The calculated value is quite close. (L.M. Thomas and J. Shanker, "Equation of State and Bulk Modulus for NaCl", Physica Status Solidi B, 189, 363 (1995)) Comment: Equation (3) written as

−−=

m

n

r

ArE

no

o 1)(

is often called Born-Landé equation for the lattice energy of a crystal (after Max Born and Alfred Landé)

1.10 Van der Waals bonding Below 24.5 K, Ne is a crystalline solid with an FCC structure. The interatomic interaction energy per atom can be written as

−

−=

126

13.1245.142)(rr

rEσσε (eV/atom)

where ε and σ are constants that depend on the polarizability, the mean dipole moment, and the extent of overlap of core electrons. For crystalline Ne, ε = 3.121 × 10−3 eV and σ = 0.274 nm.

a. Show that the equilibrium separation between the atoms in an inert gas crystal is given by ro = (1.090)σ. What is the equilibrium interatomic separation in the Ne crystal?

b. Find the bonding energy per atom in solid Ne.

c. Calculate the density of solid Ne (atomic mass = 20.18).

Solution

a. Let E = potential energy and x = distance variable between the atoms. The energy E is given by






−

−=

126

13.1245.142)(xx

xEσσε

The force F on each atom is given by

−

=−=2

5

2

11

7.8656.1452)(

)(x

x

x

x

dx

xdExF

σσσσε

∴

−=

7

6

13

12

7.8656.1452)(xx

xFσσε

When the atoms are in equilibrium, this net force must be zero. Using ro to denote equilibrium separation,

0)( =orF

∴ 07.8656.14527

6

13

12

=

−

oo rr

σσε

∴ 7

6

13

12

7.8656.145oo rr

σσ =

∴ 6

12

7

13

7.86

56.145

σσ

=

o

o

r

r

∴ ro = 1.090σ

For the Ne crystal, σ = 2.74 × 10−10 m and ε = 0.003121 eV. Therefore,

ro = 1.090(2.74 × 10−10 m) = 2.99 × 10−10 m for Ne.

b. Calculate energy per atom at equilibrium:

−

−=

126

13.1245.142)(oo

o rrrE

σσε

∴ ( )( )

××−

××

×−= −12

10-

10-

6

10-

10-

19

m 102.99

m 102.7413.12

m 102.99

m 102.7445.14

J/eV 10602.1eV 003121.02)( orE

∴ E(ro) = −4.30 × 10−21 or −0.0269 eV

Therefore the bonding energy in solid Ne is 0.027 eV per atom or 2.6 kJ / mole






c. To calculate the density, remember that the unit cell is FCC, and density = (mass of atoms in the unit cell) / (volume of unit cell). There are 4 atoms per FCC unit cell, and the atomic mass of Ne is 20.18 g/mol. (See Figure 1Q10-1)

Figure 1Q10-1: Left: An FCC unit cell with close-packed spheres. Right: Reduced-sphere representation of the FCC unit cell.

Since this is an FCC crystal structure, let a = lattice parameter (side of cubic cell) and R = radius of atom. The shortest interatomic separation is ro = 2R (atoms in contact means nucleus to nucleus separation is 2R (see Figure 1Q10-1).

R = ro/2

and 2a2 = (4R)2

∴ ( )m 1099.222

2222 10−×=

== orRa

∴ a = 4.228 × 10−10 m or 0.423 nm

The mass (mNe) of 1 Ne atom in grams is the atomic mass (Mat) divided by NA (Avogadro's number) because NA number of atoms have a mass of Mat grams,

mNe = Mat / NA

∴ ( )( )

kg 103.351mol 106.022

kg/g 001.0g/mol 18.20 261-23Ne

−×=×

=m

There are 4 atoms per unit cell in the FCC lattice. The density ρ can then be found by

ρ = (4mNe) / Volume of unit cell

or ρ = (4mNe) / a3 = [4 × (3.351 × 10−26 kg)] / (4.228 × 10−10 m)3

∴ ρ = 1774 kg/m3

In g/cm3, this density is:

ρ = ( ) ( )g/kg 1000cm/m 100

kg/m 17743

3

× = 1.77 g/cm3

The density of solid Ne is 1.77 g cm−3.

------------------------ Notes:






1. Except for He, all inert gas crystal structures are FCC.

2. The calculated value is very close to the experimental value for a (below 24 K, or −246 °C), which is 0.44 nm. (e.g. https://www.webelements.com/neon/crystal_structure.html, 22 October 2016). The values of ε and σ for Ne are from C. Kittel, Solid State Physics, Seventh Edition, John Wiley and Sons, New York, 1996; Table 4, p60. Values in SI units are given below

He Ne Ar Kr Xe

ε x 10-21 J 0.14 0.5 1.67 2.25 3.2 ε (meV) 0.874 3.121 10.424 14.045 19.975 σ (nm) 0.256 0.274 0.34 0.365 0.398

3. There is an excellent undergraduate level description and discussion of Lennard-Jones potential for two isolated atoms and atoms in the FCC crystal in Alan Walton, Three Phases of Matter, Second Edition, Clarendon Press (Oxford University Press), Oxford, 1983; pp43-47 for two isolated atoms and pp259-261 for atoms in the crystal.

1.11 Kinetic molecular theory

a. In particular Ar-ion laser tube the gas pressure due to Ar atoms is about 0.1 torr at 25 °C when the laser is off. What is the concentration of Ar atoms per cm3 at 25 °C in this laser? (760 torr = 1 atm = 1.013×105 Pa.)

b. In the He-Ne laser tube He and Ne gases are mixed and sealed. The total pressure P in the gas is given by contributions arising from He and Ne atoms P = PHe + PNe where PHe and PNe are the partial pressures of He and Ne in the gas mixture, that is, pressures due to He and Ne gasses alone,

=

V

RT

N

NP

A

HeHe and

=

V

RT

N

NP

A

NeNe

In a particular He-Ne laser tube the ratio of He and Ne atoms is 7:1, and the total pressure is about 1 torr at 22 °C. Calculate the concentrations of He and Ne atoms in the gas at 22 °C. What is the pressure at an operating temperature of 130 °C?

Solution

a. From the Kinetic molecular theory for gases, we have

RTN

NPV

A

=

where, R is the gas constant constant, T is the temperature. The number of Ar atoms per unit volume is

RT

PN

V

Nn A==Ar

We are given =××= Pa10013.1torr760

torr1.0 5P 13.33 Pa

Therefore the number of Ar atoms per unit volume nAr will be






)K27325)(molK J8.3145(

)mol10022.6(Pa)33.13(11

123

Ar +×= −−

−

n = 3.24 × 1021 m−3 or 3.24 × 1015 cm−3

b. Let nHe = NHe/V, the concentration of He atoms; nNe = NNe/V, the concentration of Ne atoms. Given that the total pressure is the sum of the pressure by He and Ne gasses

P = PHe + PNe

∴

+

=

+

=

AAAA N

RTn

N

RTn

V

RT

N

N

V

RT

N

NP NeHe

NeHe

∴

=

+

=

AAA N

RTn

N

RTn

N

RTnP NeNeNe 87

∴ RT

PNn A

8Ne =

Thus at T = 22 °C (295 K) for Ne,

)K22273)(molK J8.3145(8

)mol10022.6()torr760(

)torr1()Pa101013.1(

11

1235

Ne +

××= −−

−

n

= 4.09 × 1021 m−3 or 4.09 × 1015 cm−3.

Given that nHe is 7 times that of nNe, i.e.

nHe = 7× nNe = 2.86× 1022 m−3 or 2.86× 1016 cm−3.

At T = 130 °C (403 K), the atomic concentrations of He and Ne remain unchanged (the tube has the same

volume, neglecting the thermal expansion). Thus, the new pressure P′ and initial pressure P are related by

366.1)K22273(

)K130273(

/

/ =++=

′=

′=

′T

T

VRT

VTR

P

P

so that the new pressure P' is 1.37 torr. ____________________________________________________________________________________

1.12 Kinetic Molecular Theory Calculate the effective (rms) speeds of the He and Ne atoms in the He-Ne gas laser tube at room temperature (300 K).

Solution






Figure 1Q12-1: The gas atoms in the container are in random motion.

Figure 1Q12-2: The He-Ne gas laser.

To find the root mean square velocity (vrms) of He atoms at T = 300 K:

The atomic mass of He is (from Periodic Table) Mat = 4.0 g/mol. Remember that 1 mole has a mass of Mat grams. Then, one He atom has a mass (m) in kg given by:

( ) kg 10642.6kg/g 001.0mol 10022.6

g/mol 0.4 27123

at −− ×=×

×==

AN

Mm

From the kinetic theory (visualized in Figure 1Q12-1),

( ) kTm2

3

2

1 2rms =v

∴ ( )( )

( )kg 106.642

K 300K J 101.38133

27

1-23

rms −

−

××==

m

kTv = 1368 m/s

The root mean square velocity (vrms) of Ne atoms at T = 300 K can be found using the same method as above, changing the atomic mass to that of Ne, Mat = 20.18 g/mol. After calculations, the mass of one Ne atom is found to be 3.351 × 10−26 kg, and the root mean square velocity (vrms) of Ne is found to be vrms = 609 m/s.

------------------------ Note: Radiation emerging from the He-Ne laser tube (Figure 1Q12-2) is due to the Ne atoms emitting light, all in phase with each other, as explained in Ch. 3. When a Ne atom happens to be moving towards the observer, due to the Doppler Effect, the frequency of the laser light is higher. If a Ne atom happens to moving away from the observer, the light frequency is lower. Thus, the random motions of the gas atoms cause the emitted radiation not to be at a single frequency but over a range of frequencies due to the Doppler Effect. ________________________________________________________________________________

*1.13 Kinetic molecular theory and the Ar ion laser An argon ion laser has a laser tube that contains Ar atoms that produce the laser emission when properly excited by an electrical discharge. Suppose that the gas temperature inside the tube is 1300 °C (very hot).






a. Calculate the mean speed (vav), rms velocity (vrms = 2v ) and the rms speed (vrms,x = 2xv ) in one

particular direction of the Ar atoms in the laser tube, assuming 1300 °C. (See Example 1.11.) b. Consider a light source that is emitting waves and is moving towards an observer, somewhat like a

whistling train moving towards a passenger. If fo is the frequency of the light waves emitted at the source, then, due to the Doppler effect, the observer measures a higher frequency f that depends on the velocity vAr of the source towards to observer and the speed c of light,

+=

cff o

Ar1v

It is the Ar ions that emit the laser output light in the Ar-ion laser. The emission wavelength λo = c/fo is 514.5 nm. Calculate the wavelength λ registered by an observer for those atoms that are moving with a mean speed vav toward the observer. Those atoms that are moving away from the observer will result in a lower observed frequency because vAr will be negative. Estimate the width of the wavelengths (the difference between the longest and shortest wavelengths) emitted by the Ar ion laser. Solution

a. From Example 1.11 the mean speed is given by

m

kT

π8

av =v

m is the mass of a gas atom. T = 1300 °C = 1573 K. Atomic mass of Ar is Mat = 39.95 g mol−1, therefore the mass of the Ar atom is

123

1

mol10022.6

molg95.39−

−

×==

A

at

N

Mm = 6.634 × 10−23 g or 6.634 × 10−26 kg

Mean speed is given by

)kg10634.6(

)K2731300)(JK1038066.1(8826

123

av −

−−

×+×==

ππm

kTv

∴ vav = 913.32 m s−1. Root mean square RMS velocity is

)kg10634.6(

)K2731300)(KJ103806.1(3326

123

rms −

−−

×+×==

m

kTv

∴ vrms = 991.31 m s−1.

RMS speed alon x, vrms,x = 2xv in one particular direction is

3

sm31.991

3

1rms

rms,

−

== vv x = 572.33 m s−1.

b. First we consider the case when the source is moving towards the observer with average speed, vav, the frequency observed is






+=

cff o

Ar1v

where fo = c/λo = 3×108 m s−1/ 514.5 ×10−9 m = 5.8309 × 1014 s−1, vAr as calculated above is 912.32 m s−1. Therefore the frequency is

×

+×= −

−−

18

1114

1 sm103

sm32.9131s108309.5f = 5.830922× 1014 s−1

The corresponding wavelength is therefore, λ1 = c/f1 = 3×108 m s−1/ 5.830922× 1014 s−1 = 514.4984 nm. In the case when the emitting source is moving away from the observer, the frequency is

×

−×= −

−−

18

1114

2 sm103

sm32.9131108309.5 sf = 5.830886× 1014 s−1

The corresponding wavelength is therefore, λ2 = c/f2 = 3×108 m s−1/ 5.830886× 1014 s−1 = 514.5016 nm. The range of wavelengths observed by the observer is between 514.4984 nm and 514.5016 nm. The wavelength (spectral) width is Δλ= λ2 – λ1 = 514.5016 − 514.4984 nm = 0.0032 nm, very small.

------------------------ Note: The question asks for the change in the wavelength or the width in the emitted wavelengths. Four decimal places were kept in the calculations of frequency and wavelength because we are interested in these changes and the changes in the frequency and wavelength are small. It may be thought that we should similarly use higher accuracy in the velocity calculations and a more accurate c value etc but that's not necessary because the change in the frequency is actually 2fovAr/c:

c

fc

fc

ff oooArArAr 211

vvv=

−−

+=Δ

= 2(5.8309 × 1014 s−1)(913.32 m s−1) / (3 × 108 m s−1) = 3.55 × 109 s−1 = 3.55 GHz. Note to the Instructor: Some students are known to convert a range of frequencies to a range of wavelengths by taking Δλ = c/Δf, which is wrong. To convert a small range of frequencies Δf to a range of wavelengths Δλ, take λ = c/f and differentiate it,

2f

c

df

d −=λ

∴ )s1055.3()s108309.5(

sm103 192114

18

2−

−

−

××

×=Δ=Δ≈Δ ff

cf

df

d

o

λλ = 3.13 × 10−12 m = 0.00313 nm

very close to the above calculation of 0.0032 nm.

1.14 Heat capacity of gases Table 1.9 shows the experimental values of the molar heat capacity for a

few gases at 25 °C. Assume that we can neglect the vibrations of the atoms in the molecules. For each calculate the observed degree of freedom fobserved, that is f in Cm = f(R/2). For each find the expected fexpected by considering translational and rotational degrees of freedom only. What is your conclusion? Table 1.9 Heat capacities for some gases at room temperature at constant volume, CV in J mol−1 K−1. Gas Ar Ne Cl2 O2 N2 CO2 CH4 SF6






CV 12.5 12.7 25.6 21.0 20.8 28.8 27.4 89.0

Solution If a gas atom or molecule has f degrees of freedom, then its molar heat capacity at constant volume is given by

CV = f(R/2) = f(4.158 J mol−1 K−1).

Thus we can obtain the experimental f as

R

Cf V )measured(

observed = = 3.00

which for the Ar gas gives

158.4

5.12observed =f = 3.00

This is exactly what we expect since the Ar gas consists of single Ar atoms moving around randomly in the tank. The Ar values have been entered into Table 1Q14-1. The remainder of the gases are similarly calculated and entered into Table 1Q14-1.

First, notice that for monoatomic gases such as Ar and Ne, fobserved and fexpected agree very well.

For diatomic molecules such as Cl2, O2, N2, we expect 5 degrees of freedom (DOF), 3 translational and 2 rotational, ignoring the vibrational degrees of freedom. O2 and N2 follows the expected behavior at room temperature; but not Cl2. The latter exhibits more DOF than 5 because the Cl2 molecules can gain energy through the vibrations of the two Cl atoms held together by a bond that acts as a spring (Cl−Cl). The vibrational DOF would normally contribute 2 additional (R/2) values but at room temperature their contribution is not full. (Full contribution means R/2 per DOF)

CO2 is actually a linear molecule and should have 5 translational and rotational DOF without the vibrations. It shows nearly 7 DOF, arising from various vibraions.

Both CO2 and CH4 should have full 3 translational and 3 rotational DOF, i.e. fexpected = 6. However, their DOF is more than 6. Again there is a contribution from vibrational DOF.

SF6 should have 3 translational and 3 rotational degrees of freedom, 6 in total but it exhibits some 21 DOF. Clearly vibrations play a very important role in this gas

Table 1Q14-1 Heat capacities for some gases at room temperature at constant volume, CV in J mol−1 K−1 and observed and expected f. Gas Ar Ne Cl2 O2 N2 CO2 CH4 SF6 CV 12.5 12.7 25.6 21.0 20.8 28.9 27.4 89.0 fobserved 3.01 3.05 6.16 5.05 5.00 6.95 6.59 21.40 fexpected (translation + rotation only) 3 3 5 5 5 5 6 6

Note: Data in Table 1.9 on elements are from https://en.wikipedia.org/wiki/Heat_capacities_of_the_elements_(data_page) (23 October 2016), data on CH4, CO2 and SF6 gases from http://encyclopedia.airliquide.com/Encyclopedia.asp?GasID=41 (23 October 2016).






*1.15 Degrees of freedom in a gas molecule A monatomic molecule such as Ar has only three degrees of freedom (DOF) for motion along the three independent directions x, y, and z. In a system in which there are two independent atoms such Cl and Cl, the total number of DOF f is 6 because each atom has 3 degrees of freedom. Once we form a Cl2 molecule, the original 6 DOF in KE are partitioned as shown in Figure 1.78 The Cl2 molecule has 3 translational degrees of freedom, 2 rotational and 1 vibrational, summing to the original 6. The vibrational degree of freedom itself has KE and PE terms with each having an average of kT so that a vibrational degree of freedom actually has kT of energy rather than kT.

The PE term arises from the stretching and compression of the bond (which acts like a spring) during the vibrations. Put differently, each vibrational DOF has two subdegrees of freedom associated with KE and PE terms, each of which has (1/2)kT of energy. Let na be the number of atoms in a molecule. Then 3na is the total number of kinetic energy based DOF. There will always be 3 translational DOF for the molecule and at most 3 rotational degrees of freedom. There may be one or more vibrational DOF because there may be many ways in which the atoms in the molecule can vibrate, but there is a maximum. If frot and fvib are the rotational and vibrational DOF, then 3na = 3 + frot + fvib.

a. What is the vibrational DOF for Cl2. What is the maximum molar heat capacity at constant volume CV for Cl2? Given Table 1.9, what is the vibrational contribution?

b. What is the vibrational DOF for SF6. The molar heat capacity at constant volume for the SF6 gas

at 300 K is 89.0 J mol−1 K−1 but at 700 K, it is 141 J mol−1 K−1. How many vibrational DOF do you need to explain the observations at these two temperatures?

Figure 1.78 The partitioning of degrees of freedom in a diatomic molecule.

Solution a. The Cl2 molecule has 2 atoms, joined by a spring (bond). There are 2 rotational DOF, we neglect the rotation about the bond (as shown in Figure 1.78). Thus, the general rule for the DOF,

3na = 3 + frot + fvib

gives

fvib = 3na − 3 − frot = 3(2) − 3 − 2 = 1






Thus, the Cl2 molecule has 1 vibrational DOF, which itself has 2 "subdegrees" of freedom arising from the KE and PE terms. The molar constant volume heat capacity CV at most should be

CV = translational DOF × (R/2)

+ rotational DOF × (R/2)

+ vibrational DOF × R

∴ CV = 3(R/2) + 2(R/2) + 1R = (7/2)R = 29.1 J mol−1 K−1

Table 1.9 gives CV for Cl2 at 300 K as 25.6 J mol−1 K−1, not the expected full value of 29.1 J mol−1 K−1; less than the expected.

Without the vibrations this CV would have been 5(R/2) or 20.79 J mol−1 K−1. The difference between the measured and 20.79 J mol−1 K−1 is the vibrational contribution to CV, that is

Vibrational contribution to CV at 300 K = 25.6 − 20.79 = 4.81 J mol−1 K−1.

The full vibrational contribution would have been 1R, that is 8.316 J mol−1 K−1. Clearly, the vibrational contribution is about half its full capacity.

b. The SF6 molecule has 7 atoms. There are 3 rotational DOF. Thus, the general rule for the DOF,

3na = 3 + frot + fvib

gives

fvib = 3na − 3 − frot = 3(7) − 3 − 3 = 15

Thus, the SF6 molecule has 15 vibrational DOF, with each having 2 "subdegrees" of freedom arising from the KE and PE terms. The molar constant volume heat capacity CV at most should be

CV = translational DOF × (R/2)

+ rotational DOF × (R/2)

+ vibrational DOF × R

∴ CV = 3(R/2) + 3(R/2) + 15R = 18R = 149.7 J mol−1 K−1

Consider now the CV-contributions from translational and rotational motions only (non-vibrational contribution), represented by CV(TR)

CV(TR) = 3(R/2) + 3(R/2) = 25 J mol−1 K−1

The measured vibrational contribution CV(V) at 300 K is therefore

CV(V) = CV(Measured) − CV (TR) = 89 J mol−1 K−1 − 25 J mol−1 K−1 = 64 J mol−1 K−1

It is clear that the vibrational contribution is very strong; stronger than the translational and rotational contribution. Consider

Vibrational DOF = CV(V) / R = (64 J mol−1 K−1) / (8.3145 J mol−1 K−1) = 7.7

Recall that with the vibrational DOF, each DOF has an average energy of R, and be considered to be made up of KE and PE associated "subdegrees" of freedom; each with an energy R/2. Clearly at least half the vibrations are contributing to CV.

Now consider CV at 700 K (427 °C), which is 141 J mol−1 K−1. Thus, the vibrational contribution to the molar heat capacity is




chapter 1 safa kasap university of saskatchewan canada · 2017-12-04 · e = pe= ×Š = Š 4.23 ev...

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